# Help finding eigenvectors to simple 2x2 matrix

## Homework Statement

Find a fundamental set of real solutions of the system.
x'=[-0.5 1 ]x
[-1 -0.5]

## The Attempt at a Solution

I calculated the eigenvalues to be r1 = -0.5+i and r2 = -0.5-i
Then, attempting to calculate the eigenvectors,
I plugged the numbers into the system
[-0.5+0.5-i 1][v1] = [0]
[-1 -0.5+0.5-i ][v2] [0]
which simplifies to be
[-i 1] [v1] = [0]
[-1 -i ][v2] [0]
afer matrix multiplication, I get two formulas
-1(v1)i+(v2)=0
-1(v1)-(v2)i=0
However, I am at a loss for the next step. Usually when I am finding the eigenvector those two equations will have the same ratio. Please advise.

Your two formulas do have the same ratio...Just multiply the second equation by $i$.